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6. Composite transformations

We demonstrate, using algebra, why the order of two transformations - like scaling and translating - results in different outcomes. A combination of two or more transformations is called a composite transformation.

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Video transcript

- Did you get final approval from the director? Congratulations. (crunch) Earlier, we saw that translation and scaling don't commute. Let's see if we can get a better understanding of what's going on using some algebra. Suppose we translate by an amount of five and x and three and y. Pick a point, x zero y zero, in the image we're translating. That point goes to a point, x one, y one, given by, x one equals x zero plus five, y one equals y zero plus three. Now, suppose we scale about the origin by a factor of four. Where does x one y one go? Let's call the point it goes to, x two y two. Scaling says, x two equals four times x one, and y two equals four times y one. Substitute our expressions for x one and y one. X two equals four times x zero plus five which equals four times x zero plus four times five, which equals four times x zero plus 20. Y two is equal to four times y zero plus three which equals four times y zero plus 12. This factor in front of x and y is four. So the effective scale factor is still four. However, the effective translation amount is 20 and x and 12 and y. For comparison, let's do the operations in the opposite order. Scale first that is. X one equals four times x 0, and y one equals four times y zero. Then translate... So algebraically, x two equals x one plus five, which equals four times x zero plus five and y two equals y one plus three which equals four times y zero plus three. Clearly the blue equations aren't the same as the red equations. But in either case, we can write the results of combining scaling and translation in the form x two equals s times x zero plus t x, and y two equals s times y zero plus t y. Where t x stands for the effective, or final translation amount in x, and t y is the effective translation amount in y. When two or more transformations are combined we call it a composite transformation. In the next exercise, you'll be ask to verify that this general form for composite transformation consisting of scales and translations always holds, no matter how many scales and translations are combined, and no matter what the order.